Some More Results on Irrational Numbers –
Results.1) – Show that the negative of an irrational number is irrational.
Solution :- Let ‘x’ be irrational and if possible, let (-x) be rational.
Then, (-x) is rational
=> - (-x) is rational [ negative of rational is rational ]
=> x is rational
This contradicts the fact that, x is irrational
The contradiction arises by assuming that, -x is rational.
Thus whenever ‘x’ is irrational, then –x is irrational.
Result.2)- Show that the sum of a rational and an irrational is irrational
Solution:- Let ‘x’ is rational and ‘y’ is irrational
Then, we have to show that (x + y) is irrational.
If, possible let, (x + y) be rational, then,
(x + y) is rational and ‘x’ is rational
=> [(x + y) – x] is rational [difference of two rational is rational]
=> ‘y’ is rational
This contradicts the fact that ‘y’ is irrational
The contradiction arises by assuming that (x + y) is rational
Thus, whenever ‘x’ is rational ‘y’ is irrational, then (x + y) is irrational.
i.e., (2 + √5), (1/3 + √2), (-3/7 + √5) are all irrational.
Result.3) Show that the product of a non-zero rational with an irrational is irrational.
Solution.) Let, ‘x’ be a non-zero rational and ‘y’ be irrational
Then, we have to show that ‘xy’ is irrational.
If possible, let ‘xy’ be not irrational
Then, ‘xy’ is rational
Now, xy is rational and ‘x’ is non-zero rational
=> (xy) ÷ x is rational [quotient of two rationals is rational]
=> (xy)/x is rational
=> ‘y’ is rational.
This contradicts the fact that ‘y’ is irrational.
Since, the contradiction arises by assuming that ‘xy’ is rational, so xy is irrational. Thus, 6√7, 4√3/7, -5√7/11, etc, are all irrational.